New link


I recently stumbled on a website, Theorem of the Day. While the website design is very “retro” (read: extremely dated), the content of this website is excellent. Every month or so, the website adds a new mathematical theorem with a self-contained explanation and pictures. The mathematical difficulty varies, but seems to be aimed roughly at  undergraduates: for example, there is Brook Taylor’s proof of integration by parts.

The website is maintained by Robin Whitty, a combinatorist. While there are quite a few combinatorics-flavored theorems, the website includes contributions over a wide range of mathematics. This was a nice surprise for me, and what ultimately encouraged me to recommend the website. Go check it out!


Hidden Figures, coming to theaters 2017

math news

Hidden Figures is a new movie that’s coming out next year, telling the story of some pretty awesome mathematicians and physicists! With the popular Turing film The Imitation Game in 2014, and the independent Ramanujan film The Man Who Knew Infinity in 2015, it seems like films about mathematicians continue to spark people’s interest.

Stand and Deliver (1988) might be one of the first popular math movies that was not purely educational. Since then, Hollywood has given us Academy Award winners Good Will Hunting (1997) and A Beautiful Mind (2001), and several others such as 21 (2008). But while the other movies may have been great, in terms of cinematography, what Stand and Deliver did for math more than all of the others was bring the mathematician down to earth — show the public a glimpse of the mind and life of a math teacher and his students, and plant the idea that math was for more than just “unapproachable geniuses”. The films following Stand and Deliver tended to be portraits of these geniuses, who while being important to mathematics, but exasperated the stereotype of the otherworldly mathematician. This is why I am excited about Hidden Figures. The trailer seems to also to showcase mathematicians as people, not gods.

The protagonist of the film is Katherine Johnson, who worked at NASA and computed trajectories for the Apollo 11 space mission. In 2015 she received the Presidential Medal of Honor for her lifetime achievements. The film was originally a book of the same title, written by Margot Lee Shetterly. This article in the NY Times gives a nice summary of the topic, along with comments from Johnson, Christine Darden (another NASA researcher, portrayed in the book), and Shetterly.

Game, set, math

math news

Recently, the card game Set has been a trending topic in math circles. A breakthrough came on the topic of finding an upper bound on the number of cards one could lay down without any players finding a set. The problem is a sort of combinatorial problem (it falls under the umbrella of Ramsey theory), but was translated to the language of group theory and algebraic geometry where the authors used the “polynomial method” to get much, much better bounds than ever before, and a very, very short number of pages. See these papers for more information on the math. According to the first linked article, the popularized proof technique has already been used to solve other “cap set” problems, which tend to have applications in fields like computer science and information theory.


The reason I’m interested in the news is that I didn’t realize how many applications the “mathematics of Set” had — both inside and outside of mathematics. I have spent time playing it as a logic game with many students for simply that, that it practices “logical reasoning” which is the foundation on which mathematics is built. To learn that all along, I had been playing a game with these deep connections — and that I could have been telling the students about them! — I felt both excitement and regret.


As mathematicians we get excited about the chase of new ideas, new puzzles, new theoretical structures to explore. But there’s something really special about being able to communicate this excitement to the “layperson” in the terms that they’re already familiar with, like card games. It’s a real struggle in the field of operator algebras to find such connections, since there’s such an enormous amount of structure just built up to define the questions we want to tackle. Often times the connections I do know of seem superficial, in the sense that the applications don’t often drive the problems I’m interested in or the hypotheses I impose. Even so, and even while some of the “technical speak” might be out of grasp for Average Alex, I believe that with enough thought and perseverance I can find ways to phrase the math I do in ways that are intuitive and relate-able. Maybe not ways to get across the finer points, or give those easily-digestible-but-unrealistic-“real-world” applications. What I want is something like Set is to Ramsey theory — a way that will get at the heart of the matter.

(Pre)Calculus in Ancient Babylon

math news, Uncategorized

Check out this article in the New York Times!

Dr. Mathieu Ossendrijver from Humbolt University translated cuneiform tablets dating around 350-50 B.C., describing how Babylonian astronomers calculated the distance that the planet Jupiter traveled across the sky. They did this by recording its velocity at specific times, and then approximating the area under this graph — i.e. approximating the definite integral.

Tablet and Jupiter

Photo credit: Mathieu Ossendrijver/British Museum, via the New York Times.

If you haven’t taken calculus yet, the process is still understandable. We know that distance traveled = speed x time; for example, if you travel at 60 miles/hr for 2 hours, you have traveled 120 miles. This works if you keep your speed roughly the same. On the other hand, if your speed keeps changing, you could instead calculate this on small time intervals, and then add them all up to get the total distance traveled. That is what the Babylonians did.

We’ve known many ancient cultures — China, Greece, etc. — used versions of what we now call the definite integral to calculate areas of shapes. However, this is the earliest documented instance of where the “shape” was a velocity curve (a much more abstract object than say, a circle). Previously, the earliest instance of this we knew of was in England in the 1300s.

If your library has access to the journal Science, you can find more details in:

Ossyndrijver, Matthieu. “Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph.” Science. Vol. 351, Issue 6272, pp. 482-484.


First post


Trying to figure out the best way to start this blog was difficult, since I’ve been trying to pinpoint what audience I’m writing for–students, mathematicians, or the general public. I’ve begun to realize the answer is “all of the above”–I want to be able to elaborate on examples that I talk about in class; I want to be able to talk through the math I’m currently reading and my research; and I want to be able to have something to show my friends and family about what kinds of flavors of math exist beyond what they teach in schools. So while the level of background for the topics might change drastically from post to post, the idea is that there will be something here for everybody (and with the caveat that I can’t promise to post regularly, as a respectable blog would).

There are several places I could choose to begin: What’s my story? What got me into math? What am I up to right now? But instead I am going to start at the beginning of a common conversation I have with someone I’m meeting for the first time:

Them: Oh, you study math! What sort of math is it that you do?
Me: ……things?….

This is very representative of the first few conversations I had like this. I didn’t know how to answer since the general population has usually hazy memories of functions, and perhaps knows some calculus. This blog post is what I developed over the course of many conversations as answers to this question, with the best stick figures I can muster.

Flavor 1 (Operator algebras)

blog1bAnswer A: So you know how a line is 1-dimensional? Well there are a couple of things you can do with it: you can flip it (multiply by -1), you can stretch it out and in (also by multiplication). If you had a 2-dimensional plane, you might also be able to rotate it. All of these things go under the special name of linear transformations. I work with linear transformations, but on infinite dimensional spaces.

Answer B: (For those familiar with linear algebra) I do linear algebra on (possibly) infinite dimensional spaces.

Flavor 2 (Functional analysis)

Answer A: Functions are ways we can model different things in the world. For example, I could have a function that told me the population of bunnies blog1aon campus in a specific month, and it probably would be a curve that would go up and up over time. Some models can be more accurate than others, and you try to get functions that are closer and closer to what they should be. But notice, I just started talking about functions being close–so we have to have some notion of distance between functions. So we start thinking about functions being points in some space, and having a way to measure the distance between them. This new way of thinking about functions is an area I’m involved in.

Answer B: (For those familiar with series) The group of all power series that converge in a particular radius is an example of a group of functions that we can ask a lot of questions about as a group. We can also do things like add them together, and sometimes multiply, and using series we can also define a distance or norm. This can be a very powerful tool in studying functions we care about.

Flavor 3 (Noncommutative functions)

Answer A: Do you remember functions like x^2 or 3x+1 or xy-2? These are all blog1ccommutative, because the variables are real/complex numbers and x times y is always the same as y times x. You can always switch the order of multiplication. But there are many examples of functions out there where you can’t switch the order of multiplication — these are called noncommutative. So I study these kinds of functions, try to develop calculus for them, try to do the same sorts of things we do with normal commutative ones.

An example of this is if you’re sending a signal through radio waves or something like that, you might try to send a big block of numbers to represent a picture or a bunch of data to later be decoded. In this case, these blocks of numbers are noncommutative. So, if you want to describe how the signal might change because of signal noise/interference, you need some sort of noncommutative calculus.

Answer B: (For those familar with matrices) …and an example of a noncommutative function is one with matrices for variables. When you have matrices, A times B is not usually the same thing as B times A, so things get messy.

Three is a nice number, so I’ll stop here for now.